The optimal transport problem
asks how to move probability mass from one configuration to another
in a way that minimizes the amount of work (mass times distance) performed on each
infinitesimal piece of mass.
More formally [1]}, the problem seeks an optimal plan \(\pi ^*(x,y)\)  that describes how much mass should be transferred from position \(x\)  to \(y\)  satisfying:
\(\pi ^* =\underset{\pi }{\operatorname{arg\,min}}\iint _{\mathbb {R}^2} \Vert x - y\Vert ^p\,d\pi (x, y),\) 
